Let A be a 3 × 3 invertible matrix. If |adj (24A)| = |adj (3 adj (2A))|, then |A|² is equal to :

Solution

Maths sir · 4 steps

Peel the determinant from the outside in with two rules, and the whole equation becomes powers of 2 and 3.

1. Two rules. For a 3 × 3 matrix M, |adj M| = |M|² and |kM| = k³|M|. Use only these two, starting from the outermost operation.
2. Left side. |adj(24A)| = |24A|² = (24³|A|)² = 24⁶|A|² = 2¹⁸ · 3⁶ |A|².
3. Right side. |adj(3 adj(2A))| = |3 adj(2A)|² = (27 |adj(2A)|)². Inside, |adj(2A)| = |2A|² = (8|A|)² = 64|A|², so the right side is (27 × 64 |A|²)² = 3⁶ · 2¹² |A|⁴.

   2¹⁸ · 3⁶ |A|² = 3⁶ · 2¹² |A|⁴

4. So. A is invertible, so |A| ≠ 0 and we can divide by 3⁶ · 2¹² |A|². This gives |A|² = 2⁶, which is option (c).

The idea to keep. For an n × n matrix, |adj M| = |M|ⁿ⁻¹ and |kM| = kⁿ|M|. Apply them from the outside in, and write every number as prime powers before comparing.

Answer from the NTA final answer key. Solution by PrepWiser.